摘要：We investigate the global behavior of a cubic second order difference equation x n + 1 = A x n 3 + B x n 2 x n − 1 + C x n x n − 1 2 + D x n − 1 3 + E x n 2 + F x n x n − 1 + G x n − 1 2 + H x n + I x n − 1 + J $x_{n+1}=Ax_{n}^"+ Bx_{n}^,x_{n-1}+Cx_{n}x_{n-1}^,+Dx_{n-1}^"+Ex_{n}^, +Fx_{n}x_{n-1}+Gx_{n-1}^,+Hx_{n}+Ix_{n-1}+J$ , n = 0 , 1 , … $n=0,1,\ldots$ , with nonnegative parameters and initial conditions. We establish the relations for the local stability of equilibriums and the existence of period-two solutions. We then use this result to give global behavior results for special ranges of the parameters and determine the basins of attraction of all equilibrium points. We give a class of examples of second order difference equations with quadratic terms for which a discrete version of the 16th Hilbert problem does not hold. We also give the class of second order difference equations with quadratic terms for which the Julia set can be found explicitly and represent a planar quadratic curve.
关键词：basin of attraction ; competitive map ; global stable manifold ; monotonicity ; period-two solution
